Quantum computing in a three-dimensional device lattice

ABSTRACT

In a general aspect, information is encoded in data qubits in a three-dimensional device lattice. The data qubits reside in multiple layers of the three-dimensional device lattice, and each layer includes a respective two-dimensional device lattice. A three-dimensional color code is applied in the three-dimensional device lattice to detect errors in the data qubits residing in the multiple layers. A two-dimensional color code is applied in the two-dimensional device lattice in each respective layer to detect errors in one or more of the data qubits residing in the respective layer.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claim priority to U.S. Provisional Application No.62/168,539, filed May 29, 2015.

BACKGROUND

The following description relates to quantum computing in athree-dimensional device lattice.

Quantum computing generally involves storage or processing ofinformation in quantum mechanical states of light or matter. Informationstored in these systems can display the quantum properties of thestorage medium. These properties are different from classical Newtonianlaws of physics that govern classical computing hardware. Significantevidence shows that the quantum computing paradigm allows certainadvantages; for example, some problems can be solved by a quantumcomputer using exponentially fewer resources (e.g., time, memory size,energy) than would be used by the best known classical algorithms andcomputing systems. Error correcting codes have been proposed toimplement fault-tolerant quantum computing and improve the performanceof a quantum computing system.

DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram of an example quantum computing system.

FIG. 2 is a block diagram showing functional operating regions in anexample quantum processor cell.

FIG. 3 is a diagram of an example three-dimensional gauge color code.

FIG. 4 is a diagram of another example three-dimensional gauge colorcode.

FIG. 5 is a diagram of an example of a two-dimensional device lattice ina quantum processor cell.

FIG. 6 is a diagram of another example of a two-dimensional devicelattice in a quantum processor cell.

FIG. 7 is a diagram of an example three-dimensional device lattice.

FIG. 8 is a diagram of an example operating regime for an exampletwo-dimensional device lattice.

FIG. 9 is a diagram of an example of encoded two qubit operations vialattice surgery in a layer of an example three-dimensional devicelattice.

FIG. 10 is a diagram of an example error syndrome extraction scheme inthe example two-dimensional device lattice 800 shown in FIG. 8.

FIG. 11A is a diagram showing an example quantum circuit for X syndromeextraction.

FIG. 11B is a diagram showing an example quantum circuit for Z syndromeextraction.

FIG. 12 includes two flow charts showing example techniques fordimensional reduction of an error correcting code.

FIG. 13 is a flow chart showing an example technique for dimensionalextension.

FIG. 14 includes three flow charts showing example techniques forperforming quantum logic gates encoded in a three-dimensional operatingmode of a quantum processor cell.

FIG. 15 includes three flow charts showing example techniques forperforming quantum logic gates encoded in a two-dimensional operatingmode of a quantum processor cell.

FIG. 16 is a block diagram showing dynamic allocation of functionaloperating regions in an example quantum processor cell.

DETAILED DESCRIPTION

In some aspects of what is described here, quantum computing isperformed by operating a quantum processor cell that includes athree-dimensional device. In some instances, the quantum processor cellcan be operated in a fault-tolerant regime.

In some of the examples described here, a three-dimensional gauge colorcode (e.g., a three-dimensional gauge color code, or otherthree-dimensional color code) is applied to a three-dimensional devicelattice to process quantum information (in some cases, fault-tolerantly)in layers of the three-dimensional device lattice. Each layer of thethree-dimensional device lattice can be or include a two-dimensionaldevice lattice, and a two-dimensional color code can be applied to thetwo-dimensional device lattice to process information (in some cases,fault-tolerantly) within a single layer. In some instances, the quantumprocessor cell can switch between applying quantum error correction witha three-dimensional color code (applied to a three-dimensional devicelattice) and applying quantum error correction with two-dimensionalcolor codes (applied to two-dimensional device lattices). The switchingcan be performed, in some implementations, without interrupting thefault-tolerant operation of the quantum processor cell. In some cases,the switching is used to perform a universal set of fault-tolerant gatesin the quantum processor cell.

FIG. 1 is a schematic diagram of an example quantum computing system100. The example quantum computing system 100 shown in FIG. 1 includes acontrol system 110, a signal delivery system 106, and a quantumprocessor cell 102. A quantum computing system may include additional ordifferent features, and the components of a quantum computing system mayoperate as described with respect to FIG. 1 or in another manner.

The example quantum computing system 100 shown in FIG. 1 can performquantum computational tasks by executing quantum algorithms. In someimplementations, the quantum computing system 100 can perform quantumcomputation by storing and manipulating information within individualquantum states of a composite quantum system. For example, qubits (i.e.,quantum bits) can be stored in and represented by an effective two-levelsub-manifold of a quantum coherent physical system. Coupler devices canbe used to perform quantum logic operations on single qubits orconditional quantum logic operations on multiple qubits. In someinstances, the conditional quantum logic can be performed in a mannerthat allows large-scale entanglement within the quantum computingsystem. Control signals can manipulate the quantum states of individualqubits and the joint states of multiple qubits. In some instances,information can be read out from the composite quantum system bymeasuring the quantum states of the individual qubits.

In some implementations, the quantum computing system 100 can operateusing gate-based models for quantum computing. In some models,fault-tolerance can be achieved by applying a set of control andmeasurement operations to the qubits. For example, topological quantumerror-correction schemes can operate on a lattice ofnearest-neighbor-coupled qubits. In some instances, these and othertypes of quantum error-correcting schemes can be adapted for a two- orthree-dimensional lattice of nearest-neighbor-coupled qubits, forexample, to achieve fault-tolerant quantum computation. The lattice canallow each qubit to be independently controlled and measured withoutintroducing errors on other qubits in the lattice. Adjacent pairs ofqubits in the lattice can be addressed, for example, with two-qubit gateoperations that are capable of generating entanglement, independent ofother pairs in the lattice.

In some implementations, the quantum computing system 100 is constructedand operated according to a scalable quantum computing architecture. Forexample, in some cases, the architecture can be scaled to a large numberof qubits to achieve largescale general purpose coherent quantumcomputing. In some instances, the architecture is adaptable and canincorporate a variety of modes for each technical component. Forexample, the architecture can be adapted to incorporate different typesof qubit devices, coupler devices, readout devices, signaling devices,etc.

The example quantum processor cell 102 shown in FIG. 1 includes athree-dimensional device lattice 130. The quantum processor cell 102 maybe implemented according to the example quantum processor cell 102Ashown in FIG. 2, or in another manner. The three-dimensional devicelattice 130 includes multiple devices whose collective spatialarrangement corresponds to a three-dimensional lattice structure. Forexample, the devices may be regularly spaced apart from one another andarranged in a repeating spatial pattern within the quantum processorcell 102; the repeating spatial pattern may have translational symmetryin one or more spatial dimensions of the three-dimensional devicelattice 130. The three-dimensional device lattice 130 can be a compositeof multiple two-dimensional layers. For example, each two-dimensionallayer can extend in two horizontal directions (e.g., the x-direction andy-direction of a Cartesian coordinate system), and the two-dimensionallayers can be assembled in a vertical direction (e.g., the z-directionof a Cartesian coordinate system) to form the three-dimensional devicelattice 130. The layers of the three-dimensional device lattice 130shown in FIG. 1 can each include a two-dimensional device lattice. Forexample, each layer of the three-dimensional device lattice 130 may beimplemented according to the example two-dimensional device lattices500, 600, 800 shown in FIG. 5, 6, 8, respectively, or the layers may beimplemented in another manner.

The devices in the three-dimensional device lattice 130 can includequbit devices (e.g., data qubit devices, ancilla qubit devices), couplerdevices, readout devices and possibly other types of devices. Forinstance, quantum processor cell 102 may include data qubit devices thatare used to store and process quantum information, and ancilla qubitdevices that are used to detect errors. In the example quantum processorcell 102, the qubit devices each store a single qubit (a bit of quantuminformation), and the data qubit devices can collectively define acomputational state. The quantum processor cell 102 may also includereadout devices that selectively interact with the qubit devices todetect their quantum states. For example, the readout devices maygenerate readout signals that indicate the computational state of thequantum processor or quantum memory. The quantum processor cell 102 mayalso include coupler devices that selectively operate on individualqubits or pairs of qubits. For example, the coupler devices may produceentanglement or other multi-qubit states over two or more qubits in thequantum processor cell 102.

In some implementations, the example quantum processor cell 102 canprocess the quantum information stored in the qubits by applying controlsignals to the qubit devices or to the coupler devices housed in thequantum processor cell. The control signals can be configured to encodeinformation in the qubit devices, to process the information byperforming logical gates or other types of operations, or to extractinformation from the qubit devices. In some examples, the operations canbe expressed as single-qubit gates, two-qubit gates, or other types oflogical gates that operate on one or more qubits. A sequence ofoperations can be applied to the qubits to perform a quantum algorithm.The quantum algorithm may correspond to a computational task, a quantumerror-correction procedure, a quantum state distillation procedure, or acombination of these and other types of operations.

In the example shown in FIG. 1, the signal delivery system 106 providescommunication between the control system 110 and the quantum processorcell 102. For example, the signal delivery system 106 can receivecontrol signals (e.g., qubit control signals, readout control signals,coupler control signals, etc.) from the control system 110 and deliverthe control signals to the quantum processor cell 102. In someinstances, the signal delivery system 106 performs preprocessing, signalconditioning, or other operations to the control signals beforedelivering them to the quantum processor cell 102.

In the example quantum computing system 100 shown in FIG. 1, the controlsystem 110 controls operation of the quantum processor cell 102. Theexample control system 110 may include data processors, signalgenerators, interface components and other types of systems orsubsystems. In some cases, the control system 110 includes one or moreclassical computers or classical computing components.

In some implementations, the control system 110 includes a classicalcomputing system that executes software to compile instructions for thequantum processor cell 102. For example, the control system 110 maydecompose a quantum logic circuit into discrete control operations orsets of control operations that can be executed by the hardware in thequantum processor cell 102. In some examples, that control systemapplies a quantum logic circuit by causing the qubit devices and otherdevices in the quantum processor cell 102 to execute operations. Forinstance, the operations may correspond to single-qubit gates, two-qubitgates, qubit measurements, etc. The control system 110 can generatecontrol signals that are communicated to the quantum processor cell 102by the signal delivery system 106, and the devices in the quantumprocessor cell 102 execute the operations in response to the controlsignals. In some implementations, the control system 110 applies aquantum error correcting code to the qubit devices in the quantumprocessor cell. Quantum error correcting codes include surface codes andcolor codes.

In some aspects of operation, information is encoded in data qubitsresiding in multiple layers of the three-dimensional device lattice 130.For example, a single qubit of information may be written to, stored onor otherwise encoded in each data qubit. In some cases, to encodeinformation in the data qubits, the control system 110 sends controlsignals to the quantum processor cell 102. The control signals can eachbe addressed to individual data qubits, and can be configured to modifythe quantum states of the respective data qubits to which they areaddressed. For example, the control signals may be configured to transitthe data qubits to a particular computational state, to apply logicaloperations to the data qubits, or to otherwise encode information in thedata qubits. In some cases, information can be encoded in the dataqubits in another manner, for example, by a cooling process, a thermalprocess, etc. The information encoded in the data qubits can then beprocessed, for example, by applying a quantum circuit to the dataqubits. In some instances, quantum error correcting codes are applied todetect (and in some instances, to correct) errors that occur, forexample, as a result of noise processes acting on the data qubits.

In some examples, the control system 110 applies a two-dimensional colorcode to qubits in an individual layer of the three-dimensional devicelattice 130, for example, to detect (and in some instances, to correct)errors in one or more of the data qubits residing in the individuallayer. The two-dimensional color code can be considered a stabilizercode on a three-valent, three-colorable lattice, where data qubitsoperate at the vertices of a color code lattice. The stabilizers of thecode can be implemented as parity measurements on faces of the colorcode lattice. The syndrome measurements of the code can be performed bymeasuring the stabilizers (e.g., X-stabilizers and Z-stabilizers) acrossthe color code lattice, and the stabilizer measurements can be decodedusing classical decoding algorithms.

In some examples, the control system 110 applies a three-dimensionalcolor code to qubits in multiple layers of the three-dimensional devicelattice 130, for example, to detect (and in some instances, to correct)errors in one or more of the data qubits residing in the layers. Thethree-dimensional color code can be considered a stabilizer code on afour-valent, four-colorable lattice, where data qubits operate at thevertices of a color code lattice. The stabilizers of the code can beimplemented as parity measurements on the vertices of a cell. Thesyndrome measurements of the code can be performed by measuring thestabilizers (e.g., X-stabilizers and Z-stabilizers) across the colorcode lattice, and the stabilizer measurements can be decoded usingclassical decoding algorithms. In a three-dimensional gauge color code,for example, information from parity measurements on the faces of thecolor code lattice are used in decoding.

In some examples, the three-dimensional device lattice 130 can be usedto apply gauge color codes, non-gauge color codes or possibly othertypes of color codes. During the operation of non-gauge color codes,measurements of stabilizer operators are made using ancilla qubits anddata qubits in the three-dimensional device lattice 130. The stabilizeroperators of a non-gauge color code are operators with support on thevertices of volumes in the color code lattice. During the operation ofgauge color codes, measurements of gauge operators are made usingancilla qubits and data qubits in the three-dimensional device lattice130. The gauge operators of a gauge color code are operators withsupport on the vertices of faces in the color code lattice.

Generally, a lattice of data qubits can be considered n-valent whereeach data qubit is directly connected with n other data qubits in thelattice. In some of the examples described here, a lattice is consideredthree-valent where each data qubit is directly connected with (exactlyor at least) three other data qubits in the lattice, and a lattice isconsidered four-valent where each data qubit is directly connected with(exactly or at least) four other data qubits in the lattice. Data qubitsthat are directly connected in a lattice may be connected throughcircuitry or another type of device (e.g., an ancilla qubit device, acoupler device, or another device other than a data qubit device).

Generally, a lattice of data qubits can be considered n-colorable whereeach cell of the lattice can be colored with one of n distinct colors,and each cell of the lattice contacts only unlike-colored cells in thelattice. In some of the examples described here, a lattice is consideredthree-colorable where each cell of the lattice can be colored withexactly one of three distinct colors, and each cell in the latticecontacts only unlike-colored cells in the lattice; and a lattice isconsidered four-colorable where each cell of the lattice can be coloredwith one of four distinct colors, and each cell in the lattice contactsonly unlike-colored cells in the lattice. Cells (also called “solids”)in a lattice contact if they are adjacent in the lattice, for example,if they share an edge, vertex or face. For example, in an n-colorablelattice, none of the cells of the lattice share an edge, vertex or facewith another cell of the same color.

In some implementations, the lattice that defines a color code can bemapped to the physical device lattice to which the color code isapplied. For example, a three-dimensional color code can be defined in athree-dimensional lattice and mapped to the three-dimensional devicelattice 130. The mapping can include, for example, dividing thethree-dimensional lattice that defines the color code into multipleregions, and mapping the data qubits in each region to a physical(two-dimensional) layer of the three-dimensional device lattice 130. Thedata qubits in each region of the color code lattice (thethree-dimensional lattice that defines the color code) may, in somecases, form a two-dimensional device lattice within a physical layer ofthe three-dimensional device lattice 130. An example is described withrespect to FIGS. 3 and 5.

FIG. 2 is a block diagram showing functional operating regions in anexample quantum processor cell 102A. The example quantum processor cell102A shown in FIG. 2 includes a three-dimensional device lattice 130A,which includes multiple functional operating regions. The functionaloperating regions shown in FIG. 2 include stacks, cap regions andplanes; each stack includes multiple planes and at least one cap region.In some instances, these and other types of functional operating regionscan be dynamically allocated (and reallocated) during the operation ofthe quantum processor cell 102A, for example, as discussed with respectto FIG. 16 or in another manner. For instance, the stack 201 includes acap region 206 and three planes 210A, 210B, 210C. FIG. 2 shows twoadjacent stacks in the three-dimensional device lattice 130A. Thethree-dimensional device lattice 130A may extend to include additionalstacks, and the stacks may extend to include additional planes, capregions or other functional operation regions.

In the example shown in FIG. 2, the cap region 206 includes athree-dimensional device lattice, which is a sub-lattice of the largerthe three-dimensional device lattice 130A. The cap region 206 includesmultiple layers of devices, including data qubits, and each layerincludes a two-dimensional device lattice. In some instances, athree-dimensional color code can be applied to the three-dimensionaldevice lattice in the cap region 206 to detect and correct errors in thedata qubits residing in the three-dimensional device lattice. Forexample, the three-dimensional color codes 300, 400 described withrespect to FIGS. 3 and 4, respectively, may be applied. In someinstances, a two-dimensional color code can be applied to thetwo-dimensional lattice in individual layers within the cap region 206,to detect and correct errors in the data qubits residing in theindividual layer. For example, the two-dimensional color code describedwith respect to FIG. 8 may be applied to each layer.

In the example shown in FIG. 2, each plane 210A, 210B, 210C includes atwo-dimensional device lattice, which in this example is a sub-latticeof the larger the three-dimensional device lattice 130A. Each individualplane 210A, 210B, 210C includes a layer of devices, which include dataqubits. In some instances, a two-dimensional color code can be appliedin the two-dimensional device lattice in each respective plane 210A,210B, 210C to detect errors in one or more of the data qubits residingin the respective plane.

In some instances, a quantum logic circuit is executed by the quantumprocessor cell 102A, and the quantum logic gates in the circuit can beexecuted within individual functional operating regions or acrossmultiple functional operating regions. For instance, single-qubit gates(e.g., phase gates, not gates, etc.) can be applied to individualqubits, and two-qubit gates (e.g., controlled-phase gates,controlled-not gates, swap gates, etc.) can be applied to pairs ofqubits in the same functional operating region or to pairs of qubits inmultiple functional operating regions. In some implementations, auniversal set of quantum logic gates can be performed fault-tolerantlyin the quantum processor cell 102A. Examples of fault-tolerant universalsets of quantum logic gates are described with respect to FIGS. 14 and15.

In some implementations, the quantum processor cell 102A in operationachieves universal fault-tolerant quantum computation without magicstate distillation or long-range qubit interactions. For example, insome cases the quantum processor cell 102A can perform an encoded T-gateor other non-Clifford gate fault-tolerantly without magic statedistillation. As another example, in some cases the quantum processorcell 102A can perform a universal set of quantum logic operationsfault-tolerantly with data qubits that interact with neighboring dataqubits and ancilla qubits. In some cases, the quantum processor cell102A may achieve other advantages.

FIG. 2 shows examples of operations that can be performed acrossmultiple functional operating regions in the three-dimensional devicelattice 130A. In some instances, transversal controlled-not gates 230can be applied to a pair of qubits in planes 210A and 210B (e.g., wherethe control qubit resides in plane 210A and the target qubit resides inplane 210B). Example processes 1420, 1520 for performing thecontrolled-not gates 230 are shown in FIGS. 14 and 15. In someinstances, dimensional extension or reduction operations 220 can beapplied to plane 210A and cap region 206, for example, to transferinformation between the qubit devices in the plane 210A and the qubitdevices in the cap region 206. FIGS. 12 and 13 show example processesfor performing the dimensional extension or reduction operations 220 totransfer information between the plane 210A to the cap region 206. Insome instances, color code lattice surgery operations 240 can be appliedto functional operating regions in two distinct stacks in the quantumprocessor cell 102A, for example, to perform controlled-not gates to apair of qubits in the two functional operating regions. In someimplementations, these and other operations can be performedfault-tolerantly in the quantum processor cell 102A.

FIG. 3 is a diagram of an example three-dimensional gauge color code300. The diagram in FIG. 3 shows a lattice of data qubits (e.g., dataqubit 301) represented by small circles, with color faces (e.g., face302) between the data qubits. There are four types of color faces inthree-dimensional gauge color code 300, represented by the four types ofhatching indicated in the legend in FIG. 3 (“red” faces; “blue” faces;“green” faces; “yellow” faces). The color faces can act as the gaugeoperators for the gauge color code. In FIG. 3, respective pairs of dataqubit devices are connected by lines that form the edges in the lattice.The edges of the lattice (indicated by the connecting lines in FIG. 3)indicate couplings between adjacent data qubits, where two-qubit gatesmay be applied in some instances. The direct couplings betweenneighboring data may be implemented by circuitry, either with or withoutanother type of device (e.g., an ancilla qubit, coupler, or otherdevice) between them.

The example three-dimensional gauge color code 300 shown in FIG. 3 canbe applied to a four-colorable, four-valent, three-dimensional devicelattice. For example, the example three-dimensional gauge color code 300can be applied to a three-dimensional lattice in which the layers areimplemented according to the example two-dimensional device lattices500, 600, 800 shown in FIG. 5, 6, 8, respectively.

The qubit devices in the example three-dimensional gauge color code 300can be assigned to layers in the three-dimensional device latticeaccording to the cuts 305A, 305B, 305C, 305D shown in FIG. 3. Forexample, the qubit devices in the region 310A of the color code betweencuts 305A and 305B can be assigned to a first individual layer in athree-dimensional device lattice; the qubit devices in the region 310Bbetween cuts 305B and 305C can be assigned to a second individual layeradjacent to the first layer in the three-dimensional device lattice; andthe qubit devices in the region 310C between cuts 305C and 305D can beassigned to a third individual layer adjacent to the second layer in thethree-dimensional device lattice. In some examples, each layer of qubitdevices (e.g. a layer formed from the qubit devices in any of theregions 310A, 310B, 310C) forms a two-dimensional device lattice. Forinstance, the qubit devices in the region 310A can form the exampletwo-dimensional device lattices 500, 600, 800 shown in FIG. 5, 6, 8,respectively, or they may form another type of two-dimensional devicelattice.

In some instances, the qubit devices in the three-dimensional gaugecolor code 300 can form a cap region (e.g., the cap region 206 in FIG.2) within a three-dimensional device lattice. Accordingly, in someinstances, the three-dimensional gauge color code 300 can be applied toa functional operating region within a quantum processor cell, forexample, to the cap region 206 in the example quantum processor cell102A shown in FIG. 2 or to another type of functional operating region.

FIG. 4 is a diagram of another example three-dimensional gauge colorcode 400. The diagram in FIG. 4 shows a lattice of data qubits withcolor faces between the data qubits. The data qubits are represented byvertices of the lattice shown in FIG. 4. Similar to FIG. 3, there arefour types of color faces in three-dimensional gauge color code 400 inFIG. 4, represented by the four types of hatching in FIG. 4. The colorfaces can act as the gauge operators for the gauge color code. Similarto FIG. 3, the edges of the lattice are represented by lines in FIG. 4,which indicate direct couplings between adjacent data qubits, wheretwo-qubit gates may be applied in some instances.

The example three-dimensional gauge color code 400 shown in FIG. 4 canbe applied to a four-colorable, four-valent, three-dimensional devicelattice. In some cases, the qubit devices in the examplethree-dimensional gauge color code 400 can be assigned to layers in athree-dimensional device lattice according to the cuts 405A, 405B shownin FIG. 4. For example, the qubit devices in the region 410 of the colorcode between cuts 405A and 405B can be assigned to a first individuallayer in a three-dimensional device lattice; etc. In some examples, eachlayer of qubit devices (e.g. a layer formed from the qubit devices inthe region 410) forms a two-dimensional device lattice.

In some instances, the qubit devices in the three-dimensional gaugecolor code 400 can form a cap region (e.g., the cap region 206 in FIG.2) within a three-dimensional device lattice. Accordingly, in someinstances, the three-dimensional gauge color code 400 can be applied toa functional operating region within a quantum processor cell, forexample, to the cap region 206 in the example quantum processor cell102A shown in FIG. 2 or to another type of functional operating region.

FIG. 5 is a diagram of an example of a two-dimensional device lattice500 of a quantum processor cell. The example two-dimensional devicelattice 500 shown in FIG. 5 forms an individual layer within a largerthree-dimensional device lattice. For example, multiple copies of thetwo-dimensional device lattice 500 can be assembled in a verticaldirection (perpendicular to the two-dimensional lattice plane) to formthe three-dimensional device lattice, with each copy of thetwo-dimensional device lattice 500 forming an individual layer.

The example two-dimensional device lattice 500 shown in FIG. 5 includesancilla qubit devices, data qubit devices and connections between thevarious qubit devices. The legend 510 indicates the types of qubitdevices and connections. The ancilla qubit devices (e.g., ancilla qubitdevices 501, 502), represented as black circles in FIG. 5, are notdirectly connected with another layer in the three-dimensional devicelattice. The data qubit devices that are represented as unfilled circlesin FIG. 5 (e.g., data qubit device 503) and the data qubit devices thatare represented as x-circles (“⊗”) in FIG. 5 (e.g., data qubit device506) are directly connected with a data qubit in another layer in thethree-dimensional device lattice. In particular, if the two-dimensionaldevice lattice 500 is considered as the n-th layer in thethree-dimensional device lattice, then the data qubit represented asunfilled circles are directly connected to a data qubit in the (n−1)-thlayer in the three-dimensional device lattice, and the data qubitdevices represented as x-circles are directly connected to a data qubitin the (n+1)-th layer in the three-dimensional device lattice. The dataqubit devices that are represented as plus-circles (“⊗”) in FIG. 5(e.g., data qubit device 504) are not directly connected with a dataqubit in another layer in the three-dimensional device lattice. Thesolid lines in FIG. 5 represent connections between qubit devices thatcan be used to perform syndrome checks in the color code. The dashedlines in FIG. 5 represent connections between qubit devices that can beused to perform data operations (e.g., two-qubit quantum logic gates).In the example shown in FIG. 5, the direct data connections betweenneighboring data qubit devices do not include another type of devicebetween them.

In some implementations, the example three-dimensional gauge color code300 shown in FIG. 3 can be applied to a three-dimensional device latticeformed from multiple layers of the two-dimensional device lattice 500.In such implementations, the ancilla qubits (e.g., the ancilla qubitdevice 502) can be placed at the center of the gauge operators. In somecases, multiple copies of another type of two-dimensional device latticecan be assembled to form a three-dimensional device lattice to which theother example three-dimensional gauge color code 400 shown in FIG. 4 canbe applied.

FIG. 6 is a diagram of another example of a two-dimensional devicelattice 600 of a quantum processor cell. The example two-dimensionaldevice lattice 600 shown in FIG. 6 forms an individual layer within alarger three-dimensional device lattice. For example, multiple copies ofthe two-dimensional device lattice 600 can be assembled in a verticaldirection (perpendicular to the two-dimensional lattice plane) to formthe three-dimensional device lattice, with each copy of thetwo-dimensional device lattice 600 forming an individual layer.

The example two-dimensional device lattice 600 shown in FIG. 6 includesancilla qubit devices, data qubit devices and connections between thevarious qubit devices. The legend 610 indicates the types of qubitdevices and connections, with the same meaning as in FIG. 5.

The arrangement of the data qubit devices in the example two-dimensionaldevice lattice 600 shown in FIG. 6 is the same as the arrangement of thedata qubit devices in the example two-dimensional device lattice 500shown in FIG. 5. However, the ancilla qubit devices are arranged indifferent locations in the example two-dimensional device lattices 500and 600. In particular, the positions of the ancilla qubit devices inFIG. 5 correspond to the centers of the color faces in the examplethree-dimensional gauge color code 300 shown in FIG. 3; and thepositions of the ancilla qubit devices in FIG. 6 (e.g., ancilla qubitdevices 601, 602, 603, 604) correspond to the centers of the latticeedges in the example three-dimensional gauge color code 300 shown inFIG. 3. Accordingly, the ancilla qubit devices in FIG. 5 can operate onrespective faces of the example three-dimensional gauge color code 300shown in FIG. 3; and the ancilla qubit devices in FIG. 6 can operate onrespective edges in the example three-dimensional gauge color code 300shown in FIG. 3. The ancilla qubit devices can also be placed betweendata qubit devices in different layers of the three-dimensional devicelattice formed from multiple copies of the two-dimensional devicelattice 600. In some implementations, the ancilla qubit devices can alsoact as coupler devices for performing quantum logic operations betweenthe data qubits.

In the example shown in FIG. 6, the direct data connections betweenneighboring pairs of data qubits include ancilla qubit devices. In someinstances, an ancilla qubit device in FIG. 6 can operate as a couplerdevice, for example, to implement two-qubit gates on the pair ofneighboring data qubits that are adjacent to the ancilla qubit device.

FIG. 5 and FIG. 6. show two examples of how the ancilla qubit devicescan be arranged in two-dimensional device lattices. Other arrangementsof ancilla qubit devices can be implemented, including designs that mixancillas placed on some color faces and ancillas placed on some latticeedges, designs that place two or more ancillas on some lattice edges orcolor faces, or other combinations.

FIG. 7 is a diagram of an example three-dimensional device lattice 700.In this example, the three-dimensional device lattice 700 includes qubitdevices, some of which operate as data qubit devices 702 and othersoperate as ancilla qubit devices 701. In some implementations, thethree-dimensional device lattice 700 includes a superconducting quantumcircuit that includes the ancilla qubit devices 701, the data qubitdevices 702 and other devices. In some instances, the ancilla qubitdevices 701 may also operate as coupler devices for performing quantumlogic operations, for example, on pairs of the data qubit devices 702.

The qubit devices in the example three-dimensional device lattice 700are arranged in multiple layers, where each layer includes atwo-dimensional device lattice. In the example shown in FIG. 7, thequbit devices can be connected in-plane to other qubit devices in thesame layer through a superconducting quantum circuit defined on asubstrate that supports the qubits; some of the qubit devices are alsoconnected out-of-plane to another qubit device in a different layer, forexample, by superconducting vias that extend through a substrate betweenthe layers. The qubit devices can be connected in another manner.

Example hardware arrangements and techniques for operating amulti-dimensional device lattice are described in PCT Publication Pub.WO2015/178990, titled “Housing Qubit Devices In An ElectromagneticWaveguide System.” A three-dimensional device lattice may be implementedusing another type of hardware or another arrangement of the examplehardware shown in FIG. 7.

FIG. 8 is a diagram of an example operating regime for an exampletwo-dimensional device lattice 800. The example three-dimensional gaugecolor code 300 shown in FIG. 3 can be applied to a three-dimensionallattice formed from multiple layers of the two-dimensional devicelattice 800 in FIG. 8. As shown in FIG. 8, a two-dimensional color codecan be applied to an individual two-dimensional device lattice.

The example two-dimensional device lattice 800 shown in FIG. 8 includesancilla qubit devices, data qubit devices and connections between thevarious qubit devices. The legend 810 indicates the types of qubitdevices and connections in FIG. 8. The two-dimensional device lattice800 shown in FIG. 8 is like the two-dimensional device lattice 500 shownin FIG. 5, except that the data qubits represented as unfilled circles(e.g., data qubit 803) in FIG. 8 are directly connected with data qubitsin two other layers. In particular, if the two-dimensional devicelattice 800 is considered as the n-th layer in the three-dimensionaldevice lattice, then the data qubit devices represented as unfilledcircles in FIG. 8 are directly connected to a data qubit in the (n−1)-thlayer and to a data qubit in the (n+1)-th layer in the three-dimensionaldevice lattice. All other connections in the two-dimensional devicelattice 800 shown in FIG. 8 are the same as in the two-dimensionaldevice lattice 500 shown in FIG. 5.

FIG. 8 shows the three types of color faces in an exampletwo-dimensional gauge color code. The three types of color faces arerepresented by the three types of background in the lattice in FIG. 8.In the example shown, a first set of faces (e.g., square region 820) arerepresented by the white (unshaded) background; a second set of faces(e.g., octagonal region 810) are represented by the circle-patternedbackground; and a third set of faces (e.g., octagonal region 830) arerepresented by stippled background. The color faces can act as the gaugeoperators for the gauge color code.

In the example shown in FIG. 8, the stabilizers of the two-dimensionalcolor code are given by the three types of color faces. For example, thefirst set of faces (e.g., square region 820) can correspond to a redstabilizer; the second set of faces (e.g., octagonal region 810) cancorrespond to a blue stabilizer; and a third set of faces (e.g.,octagonal region 830) can correspond to a green stabilizer.

The example two-dimensional color code shown in FIG. 8 can be applied toqubit devices that form a three-colorable, three-valent, two-dimensionaldevice lattice. For example, the two-dimensional color code can beapplied to the subset of data qubits that have vertical connections tothe planes above and beneath them; in particular, the data qubit devicesthat are represented by unfilled circles in FIG. 8 (e.g., data qubitdevice 803) form a subset of qubits in a two-dimensional device latticeto which a two-dimensional color code can be applied. In some cases, thetwo-dimensional color code can be applied to another sub-lattice oranother subset of qubit devices within a layer of a three-dimensionaldevice lattice. In some cases, the two-dimensional color code shown inFIG. 8 can be adapted for application to qubits in the exampletwo-dimensional lattice 600 shown in FIG. 6 or another two-dimensionaldevice lattice.

FIG. 9 is a diagram of an example of encoded two qubit operations vialattice surgery in a layer of an example three-dimensional devicelattice 900. The example three-dimensional device lattice 900 includesmultiple layers; two of the layers are shown in FIG. 9. As shown in FIG.9, a first layer of the three-dimensional device lattice 900 includes atwo-dimensional device lattice 800A, and a second layer of thethree-dimensional device lattice 900 includes another two-dimensionaldevice lattice 800B. Each of the two-dimensional device lattices 800Aand 800B shown in FIG. 9 is configured according to the exampletwo-dimensional device lattice 800 shown in FIG. 8. Thethree-dimensional device lattice 900 may include additional layers, andthe layers shown in FIG. 9 may extend further to include additionaldevices.

In the example shown in FIG. 9, the data qubit devices in a first region901 of the two-dimensional device lattice 800A are used to encode afirst logical qubit, and the data qubit devices in a second region 902of the two-dimensional device lattice 800A are used to encode a secondlogical qubit. The first and second regions 901, 902 each include thedevices residing along and within their respective boundary lines shownin FIG. 9.

In some implementations, the first and second regions 901, 902 in FIG. 9can be used to encode quantum information in a two-dimensional colorcode. In some cases, the two-dimensional size of the regions 901, 902can be determined based on a required code distance. The distance of aquantum error correcting code may refer, for example, to a minimumnumber of qubits that a noise operator must apply to in order to createa logical error in the code. For example, the two-dimensional color codedescribed with respect to FIG. 8 can be implemented in the first andsecond regions 901, 902 in FIG. 9, with a code distance of five. Thebackground in and about the boundary lines in FIG. 9 corresponds to thecolor faces for the two-dimensional color code represented in FIG. 8. Inthe example shown in FIG. 9, each region 901, 902 can be used to encodea single logical qubit and the spatial position of the region within thelattice, as well as its size and code distance, can be dynamicallyallocated, for example, during the operation of a quantum processorcell.

FIG. 11A is a diagram showing an example quantum circuit 1100A for Xsyndrome extraction in a quantum error correcting code. The examplequantum circuit 1100A can be applied to an ancilla qubit and four dataqubits. In some color codes, an X syndrome extraction is applied to morethan one ancilla qubit, or an X syndrome extraction is applied to moreor fewer than four data qubits. In the example shown in FIG. 11A, anancilla qubit is prepared in the |+

state, and then controlled-not gates are applied between the ancillaqubits and each of the data qubits in the stabilizer or gauge operator.In the example shown in FIG. 11A, in each controlled-not gate, theancilla qubit acts as the control for the conditional NOT operation, andthe data qubit acts as the target for the conditional NOT operation. Theancilla qubit can then be measured in the X basis to extract the X errorsyndrome. In some quantum error correcting codes, a similar process canbe applied across multiple (e.g., all or a subset of all) ancilla qubitsin parallel.

FIG. 11B is a diagram showing an example quantum circuit 1100B for Zsyndrome extraction in a quantum error correcting code. The examplequantum circuit 1100B can be applied to an ancilla qubit and four dataqubits. In some color codes, a Z syndrome extraction is applied to morethan one ancilla qubit, or an X syndrome extraction is applied to moreor fewer than four data qubits. In the example shown in FIG. 11B, anancilla qubit is prepared in the |0

state, and then controlled-not gates are applied between the ancillaqubits and each of the data qubits in the stabilizer or gauge operator.In the example shown in FIG. 11B, in each controlled-not gate, the dataqubit acts as the control for the conditional NOT operation, and theancilla qubit acts as the target for the conditional NOT operation. Theancilla qubit can then be measured in the Z basis to extract the Z errorsyndrome. In some quantum error correcting codes, a similar process canbe applied across multiple (e.g., all or a subset of all) ancilla qubitsin parallel.

FIG. 10 is a diagram showing an example error syndrome extraction schemein the example two-dimensional device lattice 800 shown in FIG. 8. Theerror syndrome extraction scheme shown in FIG. 10 can be used for Xsyndrome extraction, Z syndrome extraction or another type of syndromeextraction operation in some cases. The example error syndromeextraction scheme represented in FIG. 10 includes six groups ofoperations that can be performed, for example, in a sequence of timesteps (e.g., six time steps). In each group of operations in the errorsyndrome extraction scheme, controlled-not gates are applied betweenpairs of qubits, analogous to the controlled-not gates shown in FIGS.11A and 11B, where each controlled-not gate is applied to an ancillaqubit and a data qubit.

The letters (A, B, C, D, E and F) along various qubit connections inFIG. 10 correspond to the six groups of operations in the error syndromeextraction scheme. In particular, the letter “A” indicates the subset ofconnections where controlled-not gates are applied in the first group ofoperations; the letter “B” indicates the subset of connections wherecontrolled-not gates are applied in the second group of operations; theletter “C” indicates the subset of connections where controlled-notgates are applied in the third group of operations; the letter “D”indicates the subset of connections where controlled-not gates areapplied in the fourth group of operations; the letter “E” indicates thesubset of connections where controlled-not gates are applied in thefifth group of operations; and the letter “F” indicates the subset ofconnections where controlled-not gates are applied in the sixth group ofoperations.

In some implementations, the groups of operations can be applied invarious sequences or in parallel. For example, all of the two-qubit gateoperations cannot necessarily be applied during a single time step. Insome instances, the six groups of operations can be scheduled over sixtime steps, with the controlled-not gates in the groups of operationsbeing performed in respective time steps. For example, in time step 1,the controlled-not gates can be applied across the connections labelled“A,” in time step 2, the controlled-not gates can be applied across theconnections labelled “B,” and so forth for the controlled-not gatesacross the connections labelled “C,” “D,” “E,” and “F.” The groups ofoperations may be performed in another order, and other schedulingschemes can be used for the error syndrome extraction scheme.

FIG. 12 includes two flow charts showing example processes 1201, 1202for dimensional reduction of an error correcting code. The exampleprocesses 1201, 1202 can be used convert a three-dimensional color codeto a two-dimensional color code on one of the faces of thethree-dimensional color code. For example, the example process 1201 canbe used to convert the three-dimensional gauge color code 300 shown inFIG. 3 to a two-dimensional color code applied to an individualtwo-dimensional face of the three-dimensional gauge color code 300.

At 1210, a three-dimensional quantum error correcting code (a “3D code”)is applied to a three-dimensional device lattice. At 1211, a target sideof the 3D code is chosen. The target side of the 3D code can be a faceof the 3D code that will be reduced to a two-dimensional quantum errorcorrecting code (the “2D code”). At 1212, a gauge syndrome is measuredon all the other data qubits in the 3D code that are not in the targetside; these data qubits can be referred to as the inner colex. Thestabilizer syndrome for the inner colex can then be recovered as itwould be recovered for the 3D code. For instance, a stabilizer for thewhole 3D code may be a product of a stabilizer for the inner colex and astabilizer for the target side. At 1213, the target side's 2D syndromeis inferred. For example, a stabilizer for the target side can berecovered as one whose product with the stabilizer of the inner colex isconsistent with a stabilizer result for the whole 3D code. At 1214, thedata qubits that are not on the target side (the qubits in the innercollex) are dropped or deallocated from the code. At 1215, atwo-dimensional quantum error correcting code (a “2D code”) is appliedto the qubits in the target side of the 3D code. In someimplementations, the example process 1201 can fault-tolerantly transferthe information from the 3D code operated at 1210 onto thetwo-dimensional target side of the 3D code.

At 1220, a three-dimensional quantum error correcting code (a “3D code”)is applied to a three-dimensional device lattice. At 1221, a target sideor face of the 3D code has been chosen for dimensional reduction. Insome examples, a property of the 3D code is that each of the fourboundary faces of the 3D code has an intrinsic color. Let the targetside that has been chosen at 1221 be colored C. All data qubits that arein the 3D code, but that are not on the target side can be referred toas the inner colex. At 1221, destructive measurements are made of allthe data qubits in the inner colex that are colored C. At 1222, theresults of this destructive measurement can be combined with stabilizerresults from the 3D code operation to infer the 2D color code syndromefor just the target face. At 1223, the qubits in the inner colex aredeallocated, and a two-dimensional quantum error correcting code (a “2Dcode”) is applied to the qubits in the target side of the 3D code. Insome implementations, the example process 1202 can fault-tolerantlytransfer the information from the 3D code operated at 1220 onto thetwo-dimensional target side of the 3D code.

FIG. 13 is a flow chart showing an example process 1300 for dimensionalextension. At 1301, a two-dimensional quantum error correcting code (a“2D code”) is applied to a two-dimensional device lattice. At 1302,neighboring qubits in other device layers are allocated to be used in athree-dimensional quantum error correcting code (a “3D code”). At 1303,stabilizer or gauge operator measurements (or both) are implemented sothat the 2D code resides on a face of the resulting 3D code. At 1304,the 3D code is operated. In some implementations, the example process1300 can fault-tolerantly transfer the information from the 2D codeoperated at 1301 into the 3D code.

FIG. 16 is a block diagram 1600 showing dynamic allocation of functionaloperating regions in an example quantum processor cell. In particular,the block diagram 1600 shows a stack, such as the stack 201 shown inFIG. 2, in three modes of operation. In the first mode of operation1601A, the stack includes a cap region 1606A and multiple planes,including three planes 1610A, 1610B and 1610C below the cap region1606A. In the second mode of operation 1601B, the stack includes two capregions 1606A and 1606B. Between the first mode of operation 1601A andthe second mode of operation 1601B, the three planes 1610A, 1610B and1610C have been reallocated to define the cap region 1606B. In the thirdmode of operation 1601C, the stack includes one cap region 1606A, andthe cap region 1606B has been reallocated to define three planes in thestack. In some implementations, the example process 1300 shown in FIG.13 can be used (e.g., between the first and second modes of operation1601A, 1601B) to fault-tolerantly transfer quantum information from theplanes 1610A, 1610B, 1610C to the cap region 1601B, and the exampleprocesses shown in FIG. 12 can be used to implement the dimensionalreduction procedure that allocates or deallocates qubits in the capregion 1606B. A different number of layers can be used to dimensionallyextend a two-dimensional code into a three-dimensional code. In someimplementations, N two-dimensional layers are used to extend to athree-dimensional code that has code distance N. In someimplementations, regions of three- or two-dimensional codes can bedynamically allocated during the course of a computation or can bespecified beforehand.

FIG. 14 includes three flow charts showing example processes 1400, 1410,1420 for performing quantum logic gates encoded in a three-dimensionaloperating mode of a quantum processor cell. In some implementations, theexample process 1400 can execute a fault-tolerant logical Hadamard gate,the example process 1410 can execute a fault-tolerant T gate, and theexample process 1420 can execute a fault-tolerant controlled-not gate.

At 1401, a three-dimensional quantum error correcting code (a “3D code”)is applied to a three-dimensional device lattice. At 1402 a Hadamardgate is applied to all data qubits in the 3D code. At 1403, the 3D codeis applied. In some implementations, the process 1401 applies afault-tolerant Hadamard gate to a logical qubit stored in the 3D code.

At 1411, a three-dimensional quantum error correcting code (a “3D code”)is applied to a three-dimensional device lattice. At 1412, a gaugefixing procedure is applied by first measuring the Z-type stabilizers ofthe 3D code. Should one of these operators return an error, an Xoperator is applied to that gauge. In the example shown, the gaugefixing procedure is fault-tolerant as it is measuring stabilizers only.At 1413, the data qubits of the 3D code are bipartitioned into sets Qand Q^(C) such that each qubit in Q is only coupled to qubits in Q^(C)and vice versa. At 1414, a rotation gate T^(k) is applied to each dataqubit in Q and a rotation T^(−k) is applied to each qubit in Q^(C),where k is a solution to k(|Q|−|Q^(C)|)≡1 mod 2n and T^(k) means krepetitions of T and T^(−k) is k repetitions of the inverse T gate. At1415, the 3D code is applied to the device lattice. In someimplementations, the process 1410 applies a fault-tolerant T gate to alogical qubit encoded in the 3D code.

In some instances, the example process 1420 may apply fault-tolerantcontrolled-not gates that start and end with logical qubits encoded ineither three-dimensional or two-dimensional codes. At 1421, athree-dimensional quantum error correcting code (a “3D code”) is appliedto a three-dimensional device lattice. At 1423, dimensional reduction isused to transfer the logical qubit to a two-dimensional code. At 1425, atransversal CNOT gate can then be applied. The transversal CNOT canapply a controlled-not gate between data qubits of two 2D codes thatoperate on different layers. At 1427, this two-dimensional code can thenbe extended back to a three-dimensional code.

At 1422, a two-dimensional quantum error correcting code (the “2D code)is applied to a two-dimensional device lattice. Fault-tolerantcontrolled-not gates can be applied transversally between neighboringlogical qubits via either a transversal CNOT or color code latticesurgery or another method. At 1425, a transversal CNOT is applied totwo-dimensional color codes on neighboring layers. At 1424 a,non-neighboring two-dimensional logical qubits can be swapped intoneighboring layers using a fault-tolerant SWAP gate comprising threeconcatenated fault-tolerant controlled-not gates. At 1424 b, the logicalqubit can be swapped back to its original location using furtherfault-tolerant SWAP gates. In some implementations, the process 1420 canbe used to apply a fault-tolerant controlled-not gate to logical qubitsin 2D and 3D codes of the device lattice.

FIG. 15 includes three flow charts showing example processes 1500, 1510,1520 for performing quantum logic gates encoded in a two-dimensionaloperating mode of a quantum processor cell. In some implementations theprocess 1500 can be used to apply a fault-tolerant Hadamard gate to alogical qubit encoded in a two-dimensional device lattice. At 1501, atwo-dimensional quantum error correcting code (a “2D code”) is appliedto a two-dimensional device lattice. At 1502, a Hadamard gate is appliedto all the data qubits in the 2D code. At 1503, the 2D code is appliedto the two-dimensional device lattice.

In some implementations, the process 1510 can apply a fault-tolerant Tgate to a logical qubit encoded in a two-dimensional device lattice. At1511, a two-dimensional code is applied to a two-dimensional devicelattice. At 1512, a two-dimensional encoded qubit can be moved, forexample, via encoded SWAP gates, near a region where qubits can beallocated for dimensional extension to a 3D code. At 1513, dimensionalextension is used to encoded the two-dimensional qubit into athree-dimensional region of the device lattice. At 1514, an encoded Tgate is applied to the three-dimensional region, for example using theprocess 1410 shown in FIG. 14 or another process. At 1515, dimensionalreduction is used to return the 3D code to a 2D code. At 1516, the 2Dcode can be optionally moved throughout the lattice using SWAP gates. At1517, the 2D code is operated in its original location in the lattice.

In some implementations, the process 1520 can be used tofault-tolerantly perform a CNOT gate between two qubits encoded indifferent 2D codes. At 1521, two two-dimensional codes are applied totwo neighboring layers of the device lattice. At 1522, a controlled-notgate is applied vertically between the two layers. At 1523, atwo-dimensional code is applied to the two-dimensional device layers.

Some of the subject matter and operations described in thisspecification can be implemented in digital electronic circuitry, or incomputer software, firmware, or hardware, including the structuresdisclosed in this specification and their structural equivalents, or incombinations of one or more of them. Some of the subject matterdescribed in this specification can be implemented as one or morecomputer programs, i.e., one or more modules of computer programinstructions, encoded on a computer storage medium for execution by, orto control the operation of, data-processing apparatus. A computerstorage medium can be, or can be included in, a computer-readablestorage device, a computer-readable storage substrate, a random orserial access memory array or device, or a combination of one or more ofthem. Moreover, while a computer storage medium is not a propagatedsignal, a computer storage medium can be a source or destination ofcomputer program instructions encoded in an artificially generatedpropagated signal. The computer storage medium can also be, or beincluded in, one or more separate physical components or media (e.g.,multiple CDs, disks, or other storage devices).

Some of the operations described in this specification can beimplemented as operations performed by a data processing apparatus ondata stored on one or more computer-readable storage devices or receivedfrom other sources.

The term “data-processing apparatus” encompasses all kinds of apparatus,devices, and machines for processing data, including by way of example aprogrammable processor, a computer, a system on a chip, or multipleones, or combinations, of the foregoing. The apparatus can includespecial purpose logic circuitry, e.g., an FPGA (field programmable gatearray) or an ASIC (application specific integrated circuit). Theapparatus can also include, in addition to hardware, code that createsan execution environment for the computer program in question, e.g.,code that constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, a cross-platform runtimeenvironment, a virtual machine, or a combination of one or more of them.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, object, orother unit suitable for use in a computing environment. A computerprogram may, but need not, correspond to a file in a file system. Aprogram can be stored in a portion of a file that holds other programsor data (e.g., one or more scripts stored in a markup languagedocument), in a single file dedicated to the program, or in multiplecoordinated files (e.g., files that store one or more modules, subprograms, or portions of code). A computer program can be deployed to beexecuted on one computer or on multiple computers that are located atone site or distributed across multiple sites and interconnected by acommunication network.

Some of the processes and logic flows described in this specificationcan be performed by one or more programmable processors executing one ormore computer programs to perform actions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andprocessors of any kind of digital computer. Generally, a processor willreceive instructions and data from a read-only memory or a random-accessmemory or both. Elements of a computer can include a processor thatperforms actions in accordance with instructions, and one or more memorydevices that store the instructions and data. A computer may alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic disks, magneto optical disks, or optical disks. However, acomputer need not have such devices. Moreover, a computer can beembedded in another device, e.g., a phone, an electronic appliance, amobile audio or video player, a game console, a Global PositioningSystem (GPS) receiver, or a portable storage device (e.g., a universalserial bus (USB) flash drive). Devices suitable for storing computerprogram instructions and data include all forms of non-volatile memory,media and memory devices, including by way of example semiconductormemory devices (e.g., EPROM, EEPROM, flash memory devices, and others),magnetic disks (e.g., internal hard disks, removable disks, and others),magneto optical disks, and CD ROM and DVD-ROM disks. In some cases, theprocessor and the memory can be supplemented by, or incorporated in,special purpose logic circuitry.

To provide for interaction with a user, operations can be implemented ona computer having a display device (e.g., a monitor, or another type ofdisplay device) for displaying information to the user and a keyboardand a pointing device (e.g., a mouse, a trackball, a tablet, a touchsensitive screen, or another type of pointing device) by which the usercan provide input to the computer. Other kinds of devices can be used toprovide for interaction with a user as well; for example, feedbackprovided to the user can be any form of sensory feedback, e.g., visualfeedback, auditory feedback, or tactile feedback; and input from theuser can be received in any form, including acoustic, speech, or tactileinput. In addition, a computer can interact with a user by sendingdocuments to and receiving documents from a device that is used by theuser; for example, by sending web pages to a web browser on a user'sclient device in response to requests received from the web browser.

A computer system may include a single computing device, or multiplecomputers that operate in proximity or generally remote from each otherand typically interact through a communication network. Examples ofcommunication networks include a local area network (“LAN”) and a widearea network (“WAN”), an inter-network (e.g., the Internet), a networkcomprising a satellite link, and peer-to-peer networks (e.g., ad hocpeer-to-peer networks). A relationship of client and server may arise byvirtue of computer programs running on the respective computers andhaving a client-server relationship to each other.

In a general aspect of the subject matter described above, quantumcomputing can be performed in a three-dimensional device lattice.

In a first example, information is encoded in data qubits in athree-dimensional device lattice. The data qubits reside in multiplelayers of the three-dimensional device lattice, and each layer includesa respective two-dimensional device lattice. A three-dimensional colorcode is applied in the three-dimensional device lattice to detect errorsin the data qubits residing in the multiple layers. A two-dimensionalcolor code is applied in the two-dimensional device lattice in eachrespective layer to detect errors in one or more of the data qubitsresiding in the respective layer.

Implementations of the first example may include one or more of thefollowing features. Applying the three-dimensional color code caninclude operating ancilla qubits. Each of the ancilla qubit devices mayoperate on a respective face or a respective edge of thethree-dimensional color code. The three-dimensional color code and thetwo-dimensional color code can be applied to operate the data qubits ina universal fault-tolerant quantum computing regime.

In a second example, a quantum computing system includes a quantumprocessor cell. The quantum processor cell includes a three-dimensionaldevice lattice having multiple layers of qubit devices. Each layerincludes a respective two-dimensional device lattice. Thethree-dimensional device lattice includes a four-colorable, four-valentarrangement of the qubit devices. The two-dimensional device lattice ofeach layer includes a three-colorable, three-valent arrangement of asubset of the qubit devices in the layer.

Implementations of the second example may include one or more of thefollowing features. The qubit devices can be data qubit devices, and thequantum processor cell can further include ancilla qubit devices. Eachof the ancilla qubit devices may be configured to operate on arespective face or edge of a three-dimensional color code. The quantumprocessor cell includes a superconducting quantum circuit that includesthe qubit devices. Each of the layers includes the same three-colorable,three-valent arrangement. The quantum computing system includes acontrol system communicably coupled to the quantum processor cell andconfigured to operate the qubit devices.

In a third example, a quantum computing system includes athree-dimensional device lattice and a control system. Thethree-dimensional device lattice includes multiple layers of data qubitdevices. Each layer includes a respective two-dimensional devicelattice. The control system is communicably coupled to the quantumprocessor cell and configured to perform operations on the data qubits.The operations include encoding information in the data qubits; applyinga three-dimensional color code in the three-dimensional device latticeto detect errors in the data qubits residing in the multiple layers; andapplying a two-dimensional color code in the two-dimensional devicelattice in each respective layer to detect errors in one or more of thedata qubits residing in the respective layer.

Implementations of the third example may include one or more of thefollowing features. The three-dimensional device lattice can include afour-colorable, four-valent arrangement of the data qubit devices. Thetwo-dimensional device lattice of each layer can include athree-colorable, three-valent arrangement of a subset of the data qubitdevices in the layer. The three-dimensional device lattice furtherincludes ancilla qubit devices. The quantum computing system can includea signal delivery system that transfers signals between thethree-dimensional device lattice and the control system. Applying thethree-dimensional color code can include operating ancilla qubit devicesin the three-dimensional device lattice. The quantum computing systemincludes a quantum processor cell that houses the three-dimensionaldevice lattice. The three-dimensional device lattice can be asub-lattice of a larger three-dimensional device lattice housed in thequantum processor cell.

While this specification contains many details, these should not beunderstood as limitations on the scope of what may be claimed, butrather as descriptions of features specific to particular examples.Certain features that are described in this specification or shown inthe drawings in the context of separate implementations can also becombined. Conversely, various features that are described or shown inthe context of a single implementation can also be implemented inmultiple embodiments separately or in any suitable subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single product or packagedinto multiple products.

A number of embodiments have been described. Nevertheless, it will beunderstood that various modifications can be made. Accordingly, otherembodiments are within the scope of the following claims.

What is claimed is:
 1. A method comprising: encoding information in dataqubits in a three-dimensional device lattice, the data qubits residingin multiple layers of the three-dimensional device lattice, each layercomprising a respective two-dimensional device lattice; by operation ofa control system, applying a three-dimensional color code in thethree-dimensional device lattice to detect errors in one or more of thedata qubits residing in the multiple layers; and by operation of thecontrol system, applying a two-dimensional color code in thetwo-dimensional device lattice in each respective layer to detect errorsin one or more of the data qubits residing in the respective layer,wherein a signal delivery system transfers signals between thethree-dimensional device lattice and the control system.
 2. The methodof claim 1, wherein applying the three-dimensional color code comprisesoperating ancilla qubits.
 3. The method of claim 2, wherein each of theancilla qubits operates on a respective face of the three-dimensionalcolor code.
 4. The method of claim 2, wherein each of the ancilla qubitsoperates on a respective edge of the three-dimensional color code. 5.The method of claim 1, wherein the three-dimensional color code and thetwo-dimensional color code are applied to operate the data qubits in auniversal fault-tolerant quantum computing regime.
 6. The method ofclaim 1, wherein each of the multiple layers comprises athree-colorable, three-valent arrangement of at least a subset of thedata qubits.
 7. The method of claim 6, wherein each of the multiplelayers comprises the same three-colorable, three-valent arrangement. 8.The method of claim 1, wherein the three-dimensional device latticecomprises a four-colorable, four-valent arrangement of the data qubits,the two-dimensional device lattice of each layer comprises athree-colorable, three-valent arrangement of at least a subset of thedata qubits in the layer, and the three-dimensional device latticefurther comprises ancilla qubits in each layer.
 9. A quantum computingsystem comprising: a quantum processor cell comprising athree-dimensional device lattice comprising multiple layers of qubitdevices, each layer comprising a respective two-dimensional devicelattice, the three-dimensional device lattice comprising afour-colorable, four-valent arrangement of the qubit devices, thetwo-dimensional device lattice of each layer comprising athree-colorable, three-valent arrangement of a subset of the qubitdevices in the layer; a control system configured to: encode informationin data qubits in the three-dimensional device lattice, the data qubitsresiding in the multiple layers; apply a three-dimensional color code inthe three-dimensional device lattice to detect errors in one or more ofthe data qubits residing in the multiple layers; and apply atwo-dimensional color code in the two-dimensional device lattice in eachrespective layer to detect errors in one or more of the data qubitsresiding in the respective layer; and a signal delivery systemconfigured to transfer signals between the three-dimensional devicelattice and the control system.
 10. The quantum computing system ofclaim 9, wherein the qubit devices comprise data qubit devices, and thequantum processor cell further comprises ancilla qubit devices.
 11. Thequantum computing system of claim 10, wherein each of the ancilla qubitdevices is configured to operate in a respective face of athree-dimensional color code.
 12. The quantum computing system of claim10, wherein each of the ancilla qubit devices is configured to operatein a respective edge of a three-dimensional color code.
 13. The quantumcomputing system of claim 9, wherein the quantum processor cellcomprises a superconducting quantum circuit that includes the qubitdevices.
 14. The quantum computing system of claim 9, wherein each ofthe layers comprises the same three-colorable, three-valent arrangement.15. A quantum computing system comprising: a three-dimensional devicelattice comprising multiple layers of data qubit devices, each layercomprising a respective two-dimensional device lattice; a control systemconfigured to: encode information in data qubit devices in thethree-dimensional device lattice; apply a three-dimensional color codein the three-dimensional device lattice to detect errors in one or moreof the data qubit devices residing in the multiple layers; and apply atwo-dimensional color code in the two-dimensional device lattice in eachrespective layer to detect errors in one or more of the data qubitdevices residing in the respective layer; and a signal delivery systemconfigured to transfer signals between the three-dimensional devicelattice and the control system.
 16. The quantum computing system ofclaim 15, wherein the three-dimensional device lattice comprises afour-colorable, four-valent arrangement of the data qubit devices, thetwo-dimensional device lattice of each layer comprises athree-colorable, three-valent arrangement of a subset of the data qubitdevices in the layer, and the three-dimensional device lattice furthercomprises ancilla qubit devices in each layer.
 17. The quantum computingsystem of claim 15, wherein the three-dimensional device latticecomprises ancilla qubit devices, and applying the three-dimensionalcolor code includes operating the ancilla qubit devices.
 18. The quantumcomputing system of claim 15, comprising a quantum processor cell,wherein the three-dimensional device lattice is housed in the quantumprocessor cell.
 19. The quantum computing system of claim 18, whereinthe three-dimensional device lattice comprises a sub-lattice of a largerthree-dimensional device lattice housed in the quantum processor cell.